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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?

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Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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This challenge is about finding the difference between numbers which have the same tens digit.

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Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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Try out this number trick. What happens with different starting numbers? What do you notice?

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This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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This task follows on from Build it Up and takes the ideas into three dimensions!

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Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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Surprise your friends with this magic square trick.

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Find the sum of all three-digit numbers each of whose digits is odd.

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This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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An investigation that gives you the opportunity to make and justify predictions.

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What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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It starts quite simple but great opportunities for number discoveries and patterns!

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Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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What happens when you round these three-digit numbers to the nearest 100?

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In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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Stop the Clock game for an adult and child. How can you make sure you always win this game?

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These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.